![Page 1: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/1.jpg)
Zero-Knowledge Proof
MUIC January 30, 2019
Wutichai Chongchitmate
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,
Bangkok, Thailand
![Page 2: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/2.jpg)
1. Zero-Knowledge Proofs
2. Applications
3. Limitations and Variants
Outline
![Page 3: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/3.jpg)
Proof
x
![Page 4: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/4.jpg)
Proof
πx Prove
![Page 5: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/5.jpg)
Proof
πx Prove Verify {Accept
Reject
![Page 6: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/6.jpg)
Proof
πx Prove Verify {Accept
Reject
→ 1
→ 0
P(x) V(x,π)
![Page 7: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/7.jpg)
Proof
πx Prove Verify {Accept
Reject
→ 1
→ 0
P(x) V(x,π)
V(x,P(x)) = 1
⇔ x is true
![Page 8: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/8.jpg)
Proof
πx Prove Verify {Accept
Reject
→ 1
→ 0
P(x) V(x,π)
![Page 9: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/9.jpg)
Zero-Knowledge Proof
x
![Page 10: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/10.jpg)
Zero-Knowledge Proof
π
x
![Page 11: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/11.jpg)
Zero-Knowledge Proof
wreason why x is true
π
x
![Page 12: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/12.jpg)
Zero-Knowledge Proof
wreason why x is true
RL(x,w) = 1
π
x
![Page 13: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/13.jpg)
Zero-Knowledge Proof
wreason why x is true
RL(x,w) = 1 Probabilistic Polynomial Time (PPT)
π
x
![Page 14: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/14.jpg)
Zero-Knowledge Proof
wreason why x is true
RL(x,w) = 1 Probabilistic Polynomial Time (PPT)
π
x
![Page 15: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/15.jpg)
Zero-Knowledge Proof
wreason why x is true
RL(x,w) = 1 Probabilistic Polynomial Time (PPT)
π
x
w
![Page 16: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/16.jpg)
Zero-Knowledge Proof
wreason why x is true
RL(x,w) = 1 Probabilistic Polynomial Time (PPT)
π
x
w
![Page 17: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/17.jpg)
Zero-Knowledge Proof
wreason why x is true
RL(x,w) = 1
w?
Probabilistic Polynomial Time (PPT)
π
x
![Page 18: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/18.jpg)
NP Relation
⊆
≠
![Page 19: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/19.jpg)
NP Relation
Relation RL X x W⊆
≠
![Page 20: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/20.jpg)
NP Relation
Relation RL X x W
Language L = {x | ∃w, (x,w) ∈ RL}
⊆
≠
![Page 21: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/21.jpg)
NP Relation
Relation RL X x W
Language L = {x | ∃w, (x,w) ∈ RL}
x
⊆
≠
![Page 22: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/22.jpg)
NP Relation
Relation RL X x W
Language L = {x | ∃w, (x,w) ∈ RL}
x
w such that (x,w) ∈ RL
⊆
≠
![Page 23: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/23.jpg)
NP Relation
Relation RL X x W
Language L = {x | ∃w, (x,w) ∈ RL}
x
Difficult
w such that (x,w) ∈ RL
⊆
≠
![Page 24: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/24.jpg)
NP Relation
Relation RL X x W
Language L = {x | ∃w, (x,w) ∈ RL}
x
Difficult
w such that (x,w) ∈ RL
(x,w)
⊆
≠
![Page 25: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/25.jpg)
NP Relation
Relation RL X x W
Language L = {x | ∃w, (x,w) ∈ RL}
x
Difficult
w such that (x,w) ∈ RL
(x,w)
{1 if (x,w) ∈ RL
0 if (x,w) ∈ RL/
⊆
≠
![Page 26: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/26.jpg)
NP Relation
Relation RL X x W
Language L = {x | ∃w, (x,w) ∈ RL}
x
Difficult
w such that (x,w) ∈ RL
(x,w)
{1 if (x,w) ∈ RL
Efficient
0 if (x,w) ∈ RL/
⊆
≠
![Page 27: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/27.jpg)
NP Relation
Relation RL X x W
Language L = {x | ∃w, (x,w) ∈ RL}
RL(x,w) = 1
x
Difficult
w such that (x,w) ∈ RL
(x,w)
{1 if (x,w) ∈ RL
Efficient
0 if (x,w) ∈ RL/
⊆
≠
![Page 28: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/28.jpg)
NP Relation
Relation RL X x W
Language L = {x | ∃w, (x,w) ∈ RL}
RL(x,w) = 1
x
Difficult
w such that (x,w) ∈ RL
(x,w)
{1 if (x,w) ∈ RL
Efficient
0 if (x,w) ∈ RL/
⊆
Assume P NP≠
![Page 29: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/29.jpg)
Interactive (Probabilistic) Proof [Goldwasser, Micali, Rackoff ’85]
P(Prover)
V(Verifier)
m1
m2
m3
mr
⋮
P ↔ V
/
x
![Page 30: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/30.jpg)
Interactive (Probabilistic) Proof [Goldwasser, Micali, Rackoff ’85]
P(Prover)
V(Verifier)
m1
m2
m3
mr
⋮{1
0
P ↔ V
/
x
![Page 31: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/31.jpg)
Interactive (Probabilistic) Proof [Goldwasser, Micali, Rackoff ’85]
P(Prover)
V(Verifier)
m1
m2
m3
mr
⋮{1
0
V(x,P(x)) = 1
⇔ x is true
P ↔ V
/
x
![Page 32: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/32.jpg)
Interactive (Probabilistic) Proof [Goldwasser, Micali, Rackoff ’85]
P(Prover)
V(Verifier)
m1
m2
m3
mr
⋮{1
0
V(x,P(x)) = 1
⇔ x is true
P ↔ V
x ∈ L/
x
![Page 33: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/33.jpg)
Interactive (Probabilistic) Proof [Goldwasser, Micali, Rackoff ’85]
P(Prover)
V(Verifier)
m1
m2
m3
mr
⋮{1
0
Completeness: x ∈ L ⇒ Pr[V → 1] > 1-ε
P ↔ V
Soundness: x ∈ L ⇒ Pr[V → 1] < ε/
x
![Page 34: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/34.jpg)
Interactive (Probabilistic) Proof [Goldwasser, Micali, Rackoff ’85]
P(Prover)
V(Verifier)
m1
m2
m3
mr
⋮{1
0
Completeness: x ∈ L ⇒ Pr[V → 1] > 1-ε
P ↔ V
Soundness: x ∈ L ⇒ Pr[V → 1] < ε/
negligible
x
![Page 35: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/35.jpg)
Interactive (Probabilistic) Proof [Goldwasser, Micali, Rackoff ’85]
P(Prover)
V(Verifier)
m1
m2
m3
mr
⋮{1
0
Completeness: x ∈ L ⇒ Pr[V → 1] > 1-ε
P ↔ V
Soundness: x ∈ L ⇒ Pr[V → 1] < ε/
negligible
∀c>0 ∀k > k0, ε(k)<1/kc
x
![Page 36: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/36.jpg)
Interactive (Probabilistic) Proof [Goldwasser, Micali, Rackoff ’85]
P(Prover)
V(Verifier)
m1
m2
m3
mr
⋮{1
0
Completeness: x ∈ L ⇒ Pr[V → 1] > 1-ε
P ↔ V
Soundness: x ∈ L ⇒ Pr[V → 1] < ε/
negligible
∀c>0 ∀k > k0, ε(k)<1/kc
security parameterx
![Page 37: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/37.jpg)
Interactive (Probabilistic) Proof [Goldwasser, Micali, Rackoff ’85]
P(Prover)
V(Verifier)
m1
m2
m3
mr
⋮{1
0
Completeness: x ∈ L ⇒ Pr[V → 1] > 1-ε
P ↔ V
Soundness: x ∈ L ⇒ Pr[V → 1] < ε/
x
![Page 38: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/38.jpg)
Example 1
![Page 39: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/39.jpg)
Example 1
![Page 40: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/40.jpg)
Example 1
?
![Page 41: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/41.jpg)
Example 1
![Page 42: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/42.jpg)
Example 1
![Page 43: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/43.jpg)
Example 1
![Page 44: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/44.jpg)
Example 1
🤨
![Page 45: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/45.jpg)
Example 1
/
![Page 46: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/46.jpg)
Example 1
/
![Page 47: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/47.jpg)
Example 1
/
![Page 48: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/48.jpg)
Example 1
/
![Page 49: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/49.jpg)
Example 1
/
⇔?
![Page 50: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/50.jpg)
Example 1
Yes
/
⇔?
![Page 51: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/51.jpg)
Example 1
x20
/
![Page 52: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/52.jpg)
Example 1
x20
/
![Page 53: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/53.jpg)
Example 1
x20
/
No⇔?
![Page 54: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/54.jpg)
Example 1
x20Completeness: x ∈ L ⇒ Pr[V → 1] = 1
/
No
![Page 55: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/55.jpg)
Example 1
x20Completeness: x ∈ L ⇒ Pr[V → 1] = 1
Soundness: x ∈ L ⇒ Pr[V → 1] = 1/220/
![Page 56: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/56.jpg)
Example 1
x20Completeness: x ∈ L ⇒ Pr[V → 1] = 1
Soundness: x ∈ L ⇒ Pr[V → 1] = 1/220
≈1/1000000
/
![Page 57: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/57.jpg)
Example 1
x20Completeness: x ∈ L ⇒ Pr[V → 1] = 1
Soundness: x ∈ L ⇒ Pr[V → 1] = 1/220
≈1/1000000
/
![Page 58: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/58.jpg)
Example 1
x20Completeness: x ∈ L ⇒ Pr[V → 1] = 1
Soundness: x ∈ L ⇒ Pr[V → 1] = 1/220
?
/
![Page 59: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/59.jpg)
Zero-Knowledge Proof
P V
m1
m2
m3
mr
⋮{1
0
{1
0
![Page 60: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/60.jpg)
Zero-Knowledge Proof
P V
m1
m2
m3
mr
⋮{1
0
P(x,w) ↔ V(x) = {1
0
![Page 61: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/61.jpg)
Zero-Knowledge Proof
P V
m1
m2
m3
mr
⋮{1
0
P(x,w) ↔ V(x) =
w?
{1
0
![Page 62: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/62.jpg)
Zero-Knowledge
P V
m1
m2
m3
mr
⋮
![Page 63: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/63.jpg)
Zero-Knowledge
P V
m1
m2
m3
mr
⋮
m1
m2
m3
mr
⋮
VP
![Page 64: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/64.jpg)
Zero-Knowledge
V
m1
m2
m3
mr
⋮
VP
![Page 65: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/65.jpg)
Zero-Knowledge
S(Simulator)
V
![Page 66: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/66.jpg)
Zero-Knowledge
S(Simulator)
V
m1
m2
m3
mr
⋮
V
![Page 67: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/67.jpg)
Zero-Knowledge
m1
m2
m3
mr
⋮
VP
S(Simulator)
V
m1
m2
m3
mr
⋮
V
≃
![Page 68: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/68.jpg)
Zero-Knowledge
m1
m2
m3
mr
⋮
VP
S(Simulator)
V
m1
m2
m3
mr
⋮
V
≃ =≃s ≃c
![Page 69: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/69.jpg)
Zero-Knowledge
m1
m2
m3
mr
⋮
VP
S(Simulator)
V
m1
m2
m3
mr
⋮
V
≃ =≃s ≃c
Efficient
![Page 70: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/70.jpg)
Zero-Knowledge
m1
m2
m3
mr
⋮
VP
S(Simulator)
V
m1
m2
m3
mr
⋮
V
≃ =≃s ≃c
Efficient
![Page 71: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/71.jpg)
Interactive Proof: (P,V) V ∈ PPT
Formal Definitions
Completeness: ∀(x,w) ∈ RL, Pr[P(x,w) ↔ V(x) = 1] > 1-ε
Soundness: ∀x ∈ L,∀P*, Pr[P*(x) ↔ V(x) = 1] < ε/
![Page 72: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/72.jpg)
Zero-Knowledge Proof [GMR85]
Interactive Proof: (P,V) V ∈ PPT
Formal Definitions
Completeness: ∀(x,w) ∈ RL, Pr[P(x,w) ↔ V(x) = 1] > 1-ε
Soundness: ∀x ∈ L,∀P*, Pr[P*(x) ↔ V(x) = 1] < ε/
Zero-Knowledge: ∀V* ∈ PPT,∃SV*∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
![Page 73: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/73.jpg)
Example 2: Graph 3-Colorability
P V
![Page 74: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/74.jpg)
Example 2: Graph 3-Colorability
P V
![Page 75: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/75.jpg)
Example 2: Graph 3-Colorability
P V
![Page 76: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/76.jpg)
Example 2: Graph 3-Colorability
P V
Commitment
![Page 77: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/77.jpg)
Example 2: Graph 3-Colorability
P V
AB
![Page 78: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/78.jpg)
Example 2: Graph 3-Colorability
P V
AB
![Page 79: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/79.jpg)
Example 2: Graph 3-Colorability
P V
repeat N times
![Page 80: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/80.jpg)
Example 2: Graph 3-Colorability
P V
repeat N times
![Page 81: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/81.jpg)
Example 2: Graph 3-Colorability
P V
repeat N times
![Page 82: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/82.jpg)
Example 2: Graph 3-Colorability
P V
BC
repeat N times
![Page 83: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/83.jpg)
Example 2: Graph 3-Colorability
P V
BC
repeat N times
![Page 84: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/84.jpg)
Example 2: Graph 3-Colorability
P V
/
![Page 85: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/85.jpg)
Example 2: Graph 3-Colorability
P V
Completeness: x ∈ L ⇒ Pr[V → 1] = 1
/
![Page 86: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/86.jpg)
Example 2: Graph 3-Colorability
P V
Completeness: x ∈ L ⇒ Pr[V → 1] = 1
/
![Page 87: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/87.jpg)
Example 2: Graph 3-Colorability
P V
CE
Completeness: x ∈ L ⇒ Pr[V → 1] = 1
/
![Page 88: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/88.jpg)
Example 2: Graph 3-Colorability
P V
Completeness: x ∈ L ⇒ Pr[V → 1] = 1
/
reject
![Page 89: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/89.jpg)
Example 2: Graph 3-Colorability
P V
Completeness: x ∈ L ⇒ Pr[V → 1] = 1
Soundness: x ∈ L ⇒ Pr[V → 1] = (|E|-1/|E|)N/
![Page 90: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/90.jpg)
Example 2: Graph 3-Colorability
P V
![Page 91: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/91.jpg)
Example 2: Graph 3-Colorability
P V
3-Colorability
![Page 92: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/92.jpg)
NP-Complete and NP Reduction
source: https://en.wikipedia.org/wiki/NP-hardness
![Page 93: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/93.jpg)
NP-Complete and NP Reduction
source: https://en.wikipedia.org/wiki/NP-hardness
Algorithm A solving NP-Hard Problem
![Page 94: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/94.jpg)
NP-Complete and NP Reduction
source: https://en.wikipedia.org/wiki/NP-hardness
Algorithm A solving NP-Hard Problem
Algorithm B solving NP Problem using A in polynomial steps
![Page 95: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/95.jpg)
NP-Complete and NP Reduction
source: https://en.wikipedia.org/wiki/NP-hardness
Algorithm A solving NP-Hard Problem
Algorithm B solving NP Problem using A in polynomial steps
• Graph Coloring• Hamiltonian Path/Cycle• Subgraph Isomorphism• Boolean Satisfiability• etc.
![Page 96: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/96.jpg)
Non-Interactive Zero-Knowledge Proof (NIZK)
P V
![Page 97: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/97.jpg)
Non-Interactive Zero-Knowledge Proof (NIZK)
P V
π
![Page 98: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/98.jpg)
Non-Interactive Zero-Knowledge Proof (NIZK)
P V
π
Common Reference String (CRS)
01110011110000110101
![Page 99: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/99.jpg)
Non-Interactive Zero-Knowledge Proof (NIZK)
P V
π
Common Reference String (CRS)
01110011110000110101
Hamiltonian Cycle Problem
![Page 100: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/100.jpg)
1. Zero-Knowledge Proofs
2. Applications
3. Limitations and Variants
Outline
![Page 101: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/101.jpg)
Secure Authentication
ID, PWD
user server
![Page 102: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/102.jpg)
Secure Authentication
ID, PWD
user server
ID, PWD
![Page 103: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/103.jpg)
Secure Authentication
ID, PWD
user server
ID, PWD
![Page 104: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/104.jpg)
Secure Authentication
ID, PWD
user server
ID, PWD
ID, PWD
![Page 105: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/105.jpg)
Secure Authentication
ID, PWD
user server
ID, PWD
ID, PWD
ID, PWD
![Page 106: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/106.jpg)
Secure Authentication
ID, PWD
user server
ID, π
![Page 107: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/107.jpg)
Multi-party Computation (MPC)
![Page 108: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/108.jpg)
Multi-party Computation (MPC)
x = 0
y = 0
majority(x,y,z)
Anonymous Voting
z = 1
![Page 109: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/109.jpg)
Multi-party Computation (MPC)
x = 0
y = 0
majority(x,y,z)
Secure against semi-honest adversaries
Anonymous Voting
z = 1
![Page 110: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/110.jpg)
Multi-party Computation (MPC)
x = 0
y = 0
majority(x,y,z)
Secure against semi-honest adversaries
Secure against malicious adversaries
Anonymous Voting
z = 1
![Page 111: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/111.jpg)
Multi-party Computation (MPC)
x = 0
y = 0
majority(x,y,z)
Secure against semi-honest adversaries
Secure against malicious adversaries
Anonymous Voting
passive
z = 1
![Page 112: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/112.jpg)
Multi-party Computation (MPC)
x = 0
y = 0
majority(x,y,z)
Secure against semi-honest adversaries
Secure against malicious adversaries
Anonymous Voting
passive
active
z = 1
![Page 113: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/113.jpg)
Cryptocurrency
A pays 1 coin to B
![Page 114: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/114.jpg)
Cryptocurrency
A pays 1 coin to B
Signed A
![Page 115: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/115.jpg)
Cryptocurrency
A pays 1 coin to B
B pays 1 coin to C
Signed A
![Page 116: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/116.jpg)
Cryptocurrency
A pays 1 coin to B
B pays 1 coin to C
Signed A
Signed B
![Page 117: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/117.jpg)
Cryptocurrency
A pays 1 coin to B
Signed A
1323498
![Page 118: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/118.jpg)
Cryptocurrency
A pays 1 coin to B
Signed A
1323498Signed B
![Page 119: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/119.jpg)
Cryptocurrency
A pays 1 coin to B
Signed A
1323498Signed B
![Page 120: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/120.jpg)
Cryptocurrency
A pays 1 coin to B
Signed A
1323498
6751323498
365
Signed B
![Page 121: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/121.jpg)
Cryptocurrency
A pays 1 coin to B
Signed A
1323498
6751323498
365
1323498π
B pays to C Signed B
Signed B
![Page 122: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/122.jpg)
Cryptocurrency
A pays 1 coin to B
Signed A
1323498
6751323498
365
1323498π
B pays to C Signed B
Signed B
∃ that opens one of the locks with serial
number 1323498
![Page 123: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/123.jpg)
Cryptocurrency
A pays 1 coin to B
Signed A
1323498
6751323498
365
1323498π
B pays to C Signed B
Signed B
∃ that opens one of the locks with serial
number 1323498
B C( , 1323498)
![Page 124: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/124.jpg)
Cryptocurrency
A pays 1 coin to B
Signed A
1323498
6751323498
365
1323498π
C pays to D Signed C
Signed B
B C( , 1323498)
![Page 125: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/125.jpg)
1. Zero-Knowledge Proofs
2. Applications
3. Limitations and Variants
Outline
![Page 126: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/126.jpg)
• Length of ZKP: number of bits communicated
• Number of rounds: assuming no setup
• Setup: Common Reference String, Tokens, Correlated Randomness, etc.
• Security Assumptions: One-way functions, Number Theory-based assumptions
Limitations
![Page 127: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/127.jpg)
Variants of Zero-Knowledge Proof
Zero-Knowledge Proof
Completeness: ∀(x,w) ∈ RL, Pr[P(x,w) ↔ V(x) = 1] > 1-ε
Soundness: ∀x ∈ L,∀P*, Pr[P*(x) ↔ V(x) = 1] < ε/
Zero-Knowledge: ∀V* ∈ PPT,∃SV*∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
![Page 128: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/128.jpg)
Zero-Knowledge
• Zero-Knowledge: View(P(w),V*)(x) ≃ SV*(x)
• Perfect Zero-Knowledge: View(P(x,w) ↔ V*(x)) ≃ SV*(x)
• Statistical Zero-Knowledge: View(P(x,w) ↔ V*(x)) ≃s SV*(x)
• Computational Zero-Knowledge: View(P(x,w) ↔ V*(x)) ≃c SV*(x)
Zero-Knowledge: ∀V* ∈ PPT,∃SV*∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
![Page 129: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/129.jpg)
Zero-Knowledge
• Zero-Knowledge: View(P(w),V*)(x) ≃ SV*(x)
• Perfect Zero-Knowledge: View(P(x,w) ↔ V*(x)) ≃ SV*(x)
• Statistical Zero-Knowledge: View(P(x,w) ↔ V*(x)) ≃s SV*(x)
• Computational Zero-Knowledge: View(P(x,w) ↔ V*(x)) ≃c SV*(x)
Zero-Knowledge: ∀V* ∈ PPT,∃SV*∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
statistically close
![Page 130: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/130.jpg)
Zero-Knowledge
• Zero-Knowledge: View(P(w),V*)(x) ≃ SV*(x)
• Perfect Zero-Knowledge: View(P(x,w) ↔ V*(x)) ≃ SV*(x)
• Statistical Zero-Knowledge: View(P(x,w) ↔ V*(x)) ≃s SV*(x)
• Computational Zero-Knowledge: View(P(x,w) ↔ V*(x)) ≃c SV*(x)
Zero-Knowledge: ∀V* ∈ PPT,∃SV*∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
computationally indistinguishable
![Page 131: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/131.jpg)
Witness Indistinguishability (WI)
• Zero-Knowledge: View(P(w),V*)(x) ≃ SV*(x)
• Witness Indistinguishable: View(P(x,w1) ↔ V*(x)) ≃c View(P(x,w2) ↔ V*(x))
Zero-Knowledge: ∀V* ∈ PPT,∃SV*∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
![Page 132: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/132.jpg)
Witness Indistinguishability (WI)
• Zero-Knowledge: View(P(w),V*)(x) ≃ SV*(x)
• Witness Indistinguishable: View(P(x,w1) ↔ V*(x)) ≃c View(P(x,w2) ↔ V*(x))
Zero-Knowledge ⇒ Witness Indistinguishable
Zero-Knowledge: ∀V* ∈ PPT,∃SV*∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
![Page 133: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/133.jpg)
Soundness
• Interactive Proof• Soundness: If x not in L, then for any P*,
Pr[<P*,V>(x) = accept] = negl
• Interactive Argument: • Soundness: If x not in L, then for any efficient P*,
Pr[<P*,V>(x) = accept] = negl
Soundness: ∀x ∈ L,∀P*, Pr[P*(x) ↔ V(x) = 1] < ε
Soundness: ∀x ∈ L,∀P*∈PPT, Pr[P*(x) ↔ V(x) = 1] < ε
/
/
![Page 134: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/134.jpg)
Soundness
• Interactive Proof• Soundness: If x not in L, then for any P*,
Pr[<P*,V>(x) = accept] = negl
• Interactive Argument: • Soundness: If x not in L, then for any efficient P*,
Pr[<P*,V>(x) = accept] = negl
Interactive Argument + ZK => Zero-Knowledge Argument
Soundness: ∀x ∈ L,∀P*, Pr[P*(x) ↔ V(x) = 1] < ε
Soundness: ∀x ∈ L,∀P*∈PPT, Pr[P*(x) ↔ V(x) = 1] < ε
/
/
![Page 135: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/135.jpg)
Proof/Argument of Knowledge
• Proof of Knowledge (PoK): there exists an efficient extractor E such that for any P*, if <P*,V>(x) = accept, then Pr[EP*(x) is a witness for x] is close to 1
• Argument of Knowledge (AoK): same but for efficient P*Argument of Knowledge (AoK): ∃E ∈ PPT,∀P* ∈ PPT,
P*(x) ↔ V(x) = 1 ⇒ Pr[EP*(x) = w and (x,w) ∈ RL] > 1-ε
Proof of Knowledge (PoK): ∃E ∈ PPT,∀P*, P*(x) ↔ V(x) = 1 ⇒ Pr[EP*(x) = w and (x,w) ∈ RL] > 1-ε
![Page 136: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/136.jpg)
Black-Box vs Non-Black-Box
Zero-Knowledge: ∀V* ∈ PPT,∃SV*∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
Zero-Knowledge: ∃S∈ PPT,∀V* ∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
![Page 137: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/137.jpg)
Black-Box vs Non-Black-Box
Zero-Knowledge: ∀V* ∈ PPT,∃SV*∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
Zero-Knowledge: ∃S∈ PPT,∀V* ∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
non-black-box
![Page 138: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/138.jpg)
Black-Box vs Non-Black-Box
Zero-Knowledge: ∀V* ∈ PPT,∃SV*∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
Zero-Knowledge: ∃S∈ PPT,∀V* ∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
non-black-box
may contain the code of V*
![Page 139: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/139.jpg)
Black-Box vs Non-Black-Box
Zero-Knowledge: ∀V* ∈ PPT,∃SV*∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
Zero-Knowledge: ∃S∈ PPT,∀V* ∈ PPT, View(P(x,w) ↔ V*(x)) ≃ SV*(x)
black-box (oracle) access
![Page 140: ZK MUIC talk - MUIC Math · Zero-Knowledge Proof MUIC January 30, 2019 Wutichai Chongchitmate Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University,](https://reader033.vdocuments.site/reader033/viewer/2022043011/5fa3cd2c6ae8d055b12da6c3/html5/thumbnails/140.jpg)
Special Cases
• Σ-protocol (3-round with weaker version of argument of knowledge)
• ZAP (2-round resettably-sound resettable witness-indistinguishable proof)
• zkSNARK (Zero-Knowledge Succinct Non-Interactive Argument of Knowledge)